The free F-restriction semigroups

We provide a geometric model for the free X -generated F -restriction semigroup in the extended signature (· , ⁺,ᵐ,λ ) ( · , + , m , λ ) , where the unary operation m maps an element a to the maximum element aᵐ a m of its σ σ -class, and the constant λ λ is the unique left identity. This model is based on a certain quotient of the Cayley graph expansion of the free monoid X^* X ∗ with respect to the extended set of generators X∪ X^*̄ X ∪ X ∗ ¯ , where the generators from X^*̄ X ∗ ¯ are in a bijection with the free monoid X^* X ∗ and serve to capture the maximum elements of σ σ -classes of the quotient. We also provide models for the free X -generated strong and perfect F -restriction semigroups in the same extended signature. The constructed models enable us to solve the word problems for all the free objects under consideration.